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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z. 203
0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) + A211809(n) = n^4.

Each sequence in the following guide counts 4-tuples

(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}.  The notation "m div" means that m divides every term of the sequence.

A211058 ... wx <= yz

A211787 ... wx <= 2yz

A211795 ... wx < 2yz

A211797 ... wx > 2yz

A211809 ... wx >= 2yz

A211812 ... wx <= 3yz

A211917 ... wx < 3yz

A211918 ... wx > 3yz

A211919 ... wx >= 3yz

A211920 ... 2wx < 3yz

A211921 ... 2wx <= 3yz

A211922 ... 2wx > 3yz

A211923 ... 2wx >= 3yz

A212019 ... wx = 2yz ..... 2 div

A212020 ... wx = 3yz ..... 2 div

A212021 ... 2wx = 3yz .... 2 div

A212047 ... wx = 4yz

A212048 ... 3wx = 4yz .... 2 div

A212049 ... wx = 5yz ..... 2 div

A212050 ... 2wx = 5yz .... 2 div

A212051 ... 3wx = 5yz .... 2 div

A212052 ... 4wx = 5yz .... 2 div

A209978 ... wx = yz + 1 .. 2 div

A212053 ... wx <= yz + 1

A212054 ... wx > yz + 1

A212055 ... wx <= yz + 2

A212056 ... wx > yz + 2

A197168 ... wx = yz + 2 .. 2 div

A061201 ... w = xyz

A212057 ... w < xyz

A212058 ... w >= xyz

A212059 ... w = xyz - 1

A212060 ... w = xyz - 2

A212061 ... wx = (yz)^2

A212062 ... w^2 = xyz

A212063 ... w^2 < xyz

A212064 ... w^2 >= xyz

A212065 ... w^2 <= xyz

A212066 ... w^2 > xyz

A212067 ... w^3 = xyz

A002623 ... w = 2x + y + z

A006918 ... w = 2x + 2y + z

A000601 ... w = x + 2y + 3z (except for initial 0s)

A212068 ... 2w = x + y + z

A212069 ... 3w = x + y + z (w = average{x,y,z})

A212088 ... 3w < x + y + z

A212089 ... 3w >= x + y + z

A212090 ... w < x + y + z

A000332 ... w >= x + y + z

A212145 ... w < 2x + y + z

A001752 ... w >= 2x + y + z

A001400 ... w = 2x +3y + 4z

A005900 ... w = -x + y + z

A192023 ... w = -x + y + z + 2

A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div

A212087 ... w^2 + x^2 = y^2 + z^2

A212092 ... w^2 < x^2 + y^2 + z^2

A212093 ... w^2 <= x^2 + y^2 + z^2

A212094 ... w^2 > x^2 + y^2 + z^2

A212095 ... w^2 >= x^2 + y^2 + z^2

A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div

A212097 ... w^3 < x^3 + y^3 + z^3

A212098 ... w^3 <= x^3 + y^3 + z^3

A212099 ... w^3 > x^3 + y^3 + z^3

A212100 ... w^3 >= x^3 + y^3 + z^3

A212101 ... wx^2 = yz^2

A212102 ... 1/w = 1/x + 1/y + 1/z

A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}

A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.

A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.

A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.

A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.

A212133 ... median(w,x,y,z) = mean(w,x,y,z)

A212134 ... median(w,x,y,z) <= mean(w,x,y,z)

A212135 ... median(w,x,y,z) > mean(w,x,y,z)

A212241 ... wx + yz > n

A212243 ... 2wx + yz = n

A212244 ... w = xyz - n

A212245 ... w = xyz - 2n

A212246 ... 2w = x + y + z - n

A212247 ... 3w = x + y + z + n

A212249 ... 3w < x + y + z + n

A212250 ... 3w >= x + y + z + n

A212251 ... 3w = x + y + z + n + 1

A212252 ... 3w = x + y + z + n + 2

A212254 ... w = x + 2y + 3z - n

A212255 ... w^2 = mean(x^2, y^2, z^2)

A212256 ... 4/w = 1/x + 1/y +1/z + 1/n

In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent.  In the list below, the same is true for expressions involving more than one relation.

A000332 ... w < x <= y < z .... C(n,4)

A000914 ... w < x <= y < z .... Stirling 1st kind

A000914 ... w < x <= y >= z ... Stirling 1st kind

A050534 ... w < x < y >= z .... tritriangular

A001296 ... w <= x <= y >= z .. 4-dim pyramidal

A006322 ... x < x > y >= z

A002418 ... w < x >= y < z

A050534 ... w < x >=y >= z

A212415 ... w < x >= y <= z

A001296 ... w < x >= y <= z

A212246 ... w <= x > y <= z

A006322 ... w <= x >= y <= z

A212501 ... w > x < y >= z

A212503 ... w < 2x and y < 2z ..... A (note below)

A212504 ... w < 2x and y > 2z ..... A

A212505 ... w < 2x and y >= 2z .... A

A212506 ... w <= 2x and y <= 2z ... A

A212507 ... w < 2x and y <= 2z .... B

A212508 ... w < 2x and y < 3z ..... C

A212509 ... w < 2x and y <= 3z .... C

A212510 ... w < 2x and y > 3z ..... C

A212511 ... w < 2x and y >= 3z .... C

A212512 ... w <= 2x and y < 3z .... C

A212513 ... w <= 2x and y <= 3z ... C

A212514 ... w <= 2x and y > 3z .... C

A212515 ... w <= 2x and y >= 3z ... C

A212516 ... w > 2x and y < 3z ..... C

A212517 ... w > 2x and y <= 3z .... C

A212518 ... w > 2x and y > 3z ..... C

A212519 ... w > 2x and y >= 3z .... C

A212520 ... w >= 2x and y < 3z .... C

A212521 ... w >= 2x and y <= 3z ... C

A212522 ... w >= 2x and y > 3z .... C

A212523 ... w + x < y + z

A212560 ... w + x <= y + z

A212561 ... w + x = 2y + 2z

A212562 ... w + x < 2y + 2z ....... B

A212563 ... w + x <= 2y + 2z ...... B

A212564 ... w + x > 2y + 2z ....... B

A212565 ... w + x >= 2y + 2z ...... B

A212566 ... w + x = 3y + 3z

A212567 ... 2w + 2x = 3y + 3z

A212570 ... |w - x| = |x - y| + |y - z|

A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div

A212572 ... |w - x| <= |x - y| + |y - z| .. B

A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div

A212574 ... |w - x| >= |x - y| + |y - z| .. B

A212575 ... 2|w - x| = |x - y| + |y - z|

A212576 ... |w - x| = 2|x - y| + 2|y - z|

A212577 ... |w - x| = 2|x - y| - |y - z|

A212578 ... 2|w - x| = |x - y| - |y - z|

A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}

A212692 ... w = |x - y| + |y - z| ............... 2 div

A212568 ... w < |x - y| + |y - z| ............... 2 div

A212573 ... w <= |x - y| + |y - z| .............. 2 div

A212574 ... w > |x - y| + |y - z|

A212575 ... w >= |x - y| + |y - z|

A212676 ... w + x = |x - y| + |y - z| ......... H

A212677 ... w + y = |x - y| + |y - z|

A212678 ... w + x + y = |x - y| + |y - z|

A006918 ... w + x + y + z = |x - y| + |y - z| . H

A212679 ... |x - y| = |y - z| ................. H

A212680 ... |x - y| = |y - z| + 1 ..............H 2 div

A212681 ... |x - y| < |y - z| ................... 2 div

A212682 ... |x - y| >= |y - z|

A212683 ... |x - y| = w + |y - z| ............... 2 div

A212684 ... |x - y| = n - w + |y - z|

A212685 ... |w - x| = w + |y - z|

A186707 ... |w - x| < w + |y - z| ... (Note D)

A212714 ... |w - x| >= w + |y - z| .......... H . 2 div

A212686 ... 2*|w - x| = n + |y - z| ............. 4 div

A212687 ... 2*|w - x| < n + |y - z| ......... B

A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div

A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div

A212690 ... 2*|w - x| <= n + |y - z| ........ B

A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div

...

In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.

  R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.

A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A

A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A

A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A

A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div

A212744 ... w=range (=max-min) ............... E

A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}

A212746 ... R is in {w,x,y,z} ................ E

A212569 ... R is not in {w,x,y,z} ............ E

A212749 ... w=R or x<R or y<R or z<R ......... A

A212750 ... w=R or x=R or y<R or z<R ......... A

A212751 ... w=R or x=R or y<R or z<R ......... A

A212752 ... w<R or x<R or y<R or z>R ......... A

A212753 ... w<R or x<R or y>R or z>R ......... D

A212754 ... w<R or x>R or y>R or z>R ......... D

A002415 ... w = x + R ........................ D

A212755 ... |w - x| = R ...................... D

A212756 ... 2w = x + R

A212757 ... 2w = R

A212758 ... w = floor(R/2)

A002413 ... w = floor((x+y+z/2))

A212759 ... w, x, y are even

A212760 ... w is even and x = y + z .......... E

A212761 ... w is odd and x and y are even .... F . 2 div

A212762 ... w and x are odd y is even ........ F . 2 div

A212763 ... w, x, y are odd .................. F

A212764 ... w, x, y are even and z is odd .... F

A030179 ... w and x are even and y and z odd

A212765 ... w is even and x,y,z are odd ...... F

A212766 ... w is even and x is odd ........... A . 2 div

A212767 ... w and x are even and w+x=y+z ..... E

A212889 ... R is even ........................ A

A212890 ... R is odd ......................... A . 2 div

A212742 ... w-x, x-y, y-z are all even ....... A

A212892 ... w-x, x-y, y-z are all odd ........ A

A212893 ... w-x, x-y, y-z have same parity ... A

A005915 ... min{|w-x|, |x-y|, |y-z|} = 0

A212894 ... min{|w-x|, |x-y|, |y-z|} = 1

A212895 ... min{|w-x|, |x-y|, |y-z|} = 2

A179824 ... min{|w-x|, |x-y|, |y-z|} > 0

A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1

A212897 ... min{|w-x|, |x-y|, |y-z|} > 1

A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2

A212899 ... min{|w-x|, |x-y|, |y-z|} > 2

A212901 ... |w-x| = |x-y| = |y-z|

A212900 ... |w-x|, |x-y|, |y-z| are distinct . G

A212902 ... |w-x| < |x-y| < |y-z| ............ G

A212903 ... |w-x| <= |x-y| <= |y-z| .......... G

A212904 ... |w-x| + |x-y| + |y-z| = n ........ H

A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H

...

Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.

B: 3,-1,-5,5,1,-3,1

C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1

D: 4,-5,0,5,-4,1

E: 1,3,-3,-3,3,1,-1

F: 1,4,-4,-6,6,4,-4,-1,1

G: 2,1,-3,-1,1,3,-1,-2,1

H: 2,1,-4,1,2,-1

REFERENCES

A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.

P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

LINKS

Bo Gyu Jeong, Table of n, a(n) for n = 0..200

EXAMPLE

a(2)=11 counts these (w,x,y,z):

(1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2)

MATHEMATICA

t = Compile[{{n, _Integer}}, Module[{s = 0},

    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},

      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];

Map[t[#] &, Range[0, 40]] (* A211795 *)

(* Peter J. C. Moses, Apr 13 2012 *)

CROSSREFS

Cf. A210000, A212959.

Sequence in context: A048366 A107425 A211921 * A256226 A073720 A257364

Adjacent sequences:  A211792 A211793 A211794 * A211796 A211797 A211798

KEYWORD

nonn

AUTHOR

Clark Kimberling, Apr 27 2012

STATUS

approved

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Last modified July 26 00:44 EDT 2017. Contains 289798 sequences.